Seminar on Applicable Algebraic Geometry

2007 IMA PI Summer Program for Graduate Students
Applicable Algebraic Geometry
Seminar on Applicable Algebraic Geometry
July 23-August 10, 2007
Schedule and Abstracts
Summer Program Schedule

Schedule

Tuesday, 24 July	James Ruffo	An introduction to the Schubert calculus
Wednesday, 25 July	Hal Schenck	Multivariate polynomial splines
Thursday, 26 July	John Keyser	Algebraic Computation for Robust Geometric Modeling
Friday, 27 July	J. Maurice Rojas	P=?NP, Optimization, and Algebraic Geometry
Tuesday, 31 July	Luis Garcia	Bezier curves and surfaces
Wednesday, 1 August	Laura Matusevich	Algebraic geometry and special functions
Thursday, 2 August	Frank Sottile	Gale dual systems and bounds in real algebraic geometry
Friday, 3 August	Josephine Yu	Tropical implicitization and elimination
Monday, 6 August	Darbha Swaroop	Synthesis of fixed structure controllers using linear programming techniques
Tuesday, 7 August	Thorsten Theobald	Symmetries in SDP-based relaxations for constrained polynomial optimization
Wednesday, 8 August	Serkan Hosţen	Gröbner bases, integer programming, and disclosure limitation
Thursday, 9 August	Seth Sullivant	"Guess and check" for initial ideals

Abstracts

24.7	James Ruffo	An introduction to the Schubert calculus
	Abstract: The Schubert calculus concerns enumerative problems involving linear objects -- for instance, to count the lines in space meeting four given lines. Over the complex numbers, Schubert provided the answer to this (and many other related) problems in 1891. What might be said over the real numbers is of much recent interest. The goal of this talk is to convey the basic notions and some recent developments in this very active and fruitful subject.

25.7	Hal Schenck	Multivariate polynomial splines
	Abstract: In mathematics it is often useful to approximate a function f on a region D in R² by a ``simpler'' function. A natural way to do this is to divide D into triangles, and then approximate f on each triangle by a polynomial function. A C^r-differentiable piecewise polynomial function on D is called a spline. Splines play a key role in geometric modeling, the finite element method for solving PDE's, and in approximation theory. There is also a great deal of interesting mathematical structure involved, including commutative and homological algebra, geometry, combinatorics and topology. I'll give a gentle overview of the problem, with main focus an introduction to simplicial homology, and discuss how Grobner bases and computational algebraic geometry enter the picture.

26.7	John Keyser	Algebraic Computation for Robust Geometric Modeling
	Abstract: Many computational problems in geometric and solid modeling can be reduced into relatively easily stated algebraic formulations. As people have sought to improve the robustness of these geometric operations, greater attention has been focused on these algebraic formulations, and on methods for precisely solving such problems. In this talk, I'll begin with a brief introduction to some of the typical algebraic problems faced in geometric and solid modeling. I'll focus much of the discussion on the boundary evaluation problem for computing results of Boolean combination of solid models, and how exact computation is used to deal with these. Finally, I'll talk about some of the challenges faced when moving from more theoretical descriptions of algorithms into practical implementations.

27.7	J. Maurice Rojas	P=?NP, Optimization, and Algebraic Geometry
	Abstract: The P=?NP problem has been with us since 1971 and remains a guiding force behind theoretical computer science. The question involves a complexity class, called NP, which is comprised of hundreds of algorithms from discrete optimization, number theory, and many other areas. We give a gentle introduction to this question and then move to recent algebraic characterizations of the P=?NP problem. In particular, we will see how primes in arithmetic progression, complex roots of sparse polynomials in one variable, and circuit complexity are all brought together via complexity theory. We assume no background in complexity theory, algebraic geometry, or number theory.

31.7	Luis Garcia	Bezier curves and surfaces
	Abstract: I will introduce some basic notions in geometric modeling, which is the science of modeling curves and surfaces by small patches (e.g. Bézier patches). I will discuss some of the properties of these patches and then focus on the property known as linear precision. I will show how algebraic geometry can be used to answer some questions about linear precision for a general class of multi-sided patches known as toric patches. Finally, I will also discuss some interactions between these topics and algebraic statistics.

01.8	Laura Matusevich	Algebraic geometry and special functions
	Abstract: I will explain how commutative-algebraic knowledge of binomial ideals (namely, their primary decomposition) can be used in the study of hypergeometric functions. The lecture will include a brief overview of a combinatorial description of the primary components of binomial ideals, and requires no background in special functions or differential equations. This material is extracted from joint projects with Alicia Dickenstein, Timur Sadykov and Ezra Miller.

02.8	Frank Sottile	Gale dual systems and bounds in real algebraic geometry
	Abstract: Gale duality for polynomial systems is an elementary reformulation of a system of polynomial equations as a system of equations involving rational master functions in the complement of a hyperplane arrangement. Some properties of the original system are easier to understand in the Gale dual system. In the first part of this talk, I will describe this Gale duality, look at some examples of this construction, and give some elementary consequences. In the second half, I will explain how to bound the number of real solutions to the Gale dual system, and thereby obtain the current best bounds for the number of real solutions to a system of polynomial equations. I will also explain how this gives rise to a new numerical method for finding all real solutions to a system of polynomial equations.

03.8	Josephine Yu	Tropical implicitization and elimination
	Abstract: Implicitization is the problem of computing the defining ideal of the image of a polynomial map, and elimination is the problem of computing the defining ideal of the projection of an algebraic variety. In this talk, we will apply tropical methods to these problems. For generic cases, we can use polyhedral computations to construct the tropical varieties of the unknown ideals without computing the ideal. We will also see a demonstration of tropical softwares Gfan and TrIm.

06.8	Darbha Swaroop	Synthesis of fixed structure controllers using linear programming techniques
	Abstract: The problem of synthesizing a set of fixed structure controllers has been open for the past four decades. The problem may be simply stated as follows: Given a characteristic polynomial, P(s,K), whose coefficients are affine in the controller parameter vector, K, what values of K render P(s,K) Hurwitz? Through the Routh-Hurwitz theorem, this problem is equivalent to the determination of a set of controller parameters that satisfy a system of polynomial inequalities. In this talk, I will provide a way to approximate the set arbitrarily, at the expense of additional computation, through the use of Descartes' Rule of signs and its generalization, Hermite-Biehler Theorem and linear programming technique. I will also provide examples of the presented method to some problems encountered in engineering applications. This work is based on the doctoral dissertation of my graduate student, Waqar Malik. It is joint work with Professor S. P. Bhattacharyya of EE department at TAMU.

07.8	Thorsten Theobald	Symmetries in SDP-based relaxations for constrained polynomial optimization
	Abstract: We consider the issue of exploiting symmetries in the hierarchy of semidefinite programming relaxations recently introduced in polynomial optimization. After providing the necessary background we focus on problems where either the symmetric or the cyclic group is acting on the variables and extend the representation-theoretical methods of Gatermann and Parrilo to constrained polynomial optimization problems. Moreover, we also propose methods to efficiently compute lower and upper bounds for the subclass of problems where the objective function and the constraints are described in terms of power sums. (Joint work with L. Jansson, J.B. Lasserre and C. Riener)

08.8	Serkan Hosţen	Gröbner bases, integer programming, and disclosure limitation
	Abstract:

09.8	Seth Sullivant	"Guess and check" for initial ideals
	Abstract: A useful strategy in combinatorial commutative algebra is to "reduce to the monomial case". Generally speaking, we have an operation which takes as input an arbitrary ideal I and outputs some other ideal f(I). In the situations of interest, it might be difficult to compute f(I) directly, so we can instead try to do the same operation for an initial monomial ideal. We then cross our fingers and hope that in(f(I)) = f(in(I)) and that this operation can be used to compute f(I). In particular, in many situations, f(in(I)) proves a convenient guess for in(f(I)) from which many properties of f(I) can be deduced, including its Grobner basis. In this talk, I will try to describe two or three situations where this proves to be a useful strategy, in particular, for studying secant varieties, symbolic powers, and toric fiber products. In these cases computing f(in(I)) leads to combinatorics that is interesting in its own right. Time permitting, I will show how these results connect back to the algebraic statistical models described in my lectures.

This summer graduate program is supported by the Participating Institutions of the Institute for Mathematics and its Applications, by the US National Science Foundation, and by the Texas A&M University.

Modified since: 18 July 2007.